
with \(A_4=A_4^0+A_4^2k^2, A_3=A_3^0+A_3^2k^2, A_2=A_2^2k^2+A_2^4k^4, A_1=A_1^2k^2+A_1^4k^4, A_0=A_0^4k^4+A_0^6k^6\) .
Eq. (REF ) agrees with result obtained in [1]}, where the coefficient \(A\) 's are functions of \(\nu _{1,2,3},\varepsilon _{1,2,3},\pi _{1,2,3},\theta _{1,3},\gamma _{1,3}\)  defined earlier
(the detailed analysis will be reported elsewhere). Eq. (REF ) cannot be solved analytically and hence we present results for \(k\rightarrow 0\)  limit. At this limit,
Eq. (REF ) gives three hydrodynamic modes as, \(\omega _6^L=i\hat{h}^2\frac{k_nT}{(\epsilon _0+P_0)}k^2\)  and \(\omega _{4,5}^L= \pm c_s k+i\frac{\Gamma _s}{2} k^2+{\cal {O}}(k^3)\) ,
with scaled enthalpy per particle \(\hat{h}=(\epsilon _0+P_0)/n_0T\) ,
velocity of sound squared \(c_s^2=\big (\frac{\partial P_0}{\partial \epsilon _0}\big )_{n_0}+\frac{1}{\hat{h}}\frac{1}{T}\big (\frac{\partial P_0}{\partial n_0}\big )_{\epsilon _0}\) 
and sound attenuation coefficients \(\Gamma _s=\left[\frac{4}{3}\eta +\zeta +\frac{k_nT}{c_s^2}\big (\frac{1}{T}\frac{\partial P_0}{\partial n_0}\big )^2_{\epsilon _0}\right]/(\epsilon _0+P_0)\) .
\(\omega _6^L\)  and \(\omega _{4,5}^L\)  are the conventional heat-diffusion and sound modes of the NS theory respectively.
